Positivity of bid-ask spreads and symmetrical monotone risk aversion*

A usual argument in finance refers to no arbitrage opportunities for the positivity of the bid-ask spread. Here we follow the decision theory approach and show that if positivity of the bid-ask spread is identified with strong risk aversion for an expected utility market-maker, this is no longer true for a rank-dependent expected utility one. For such a decision-maker only a very weak form of risk aversion is required, a result which seems more in accordance with his actual behavior. We conclude by showing that the no-trade interval result of Dow and Werlang (1992a) remains valid for a rank-dependent expected utility market-maker merely exhibiting this weak form of risk aversion.     

Lottery Dependent Utility: a Reexamination

In order to accommodate empirically observed violations of the independence axiom of expected utility theory Becker and Sarin (1987) proposed their model of lottery dependent utility in which the utility of an outcome may depend on the lottery being evaluated. Although this dependence is intuitively very appealing and provides a simple functional form of the resulting decision criterion, lottery dependent utility has been nearly completely neglected in the recent literature on decision making under risk. The goal of this paper is to revive the lottery dependent utility model. Therefore, we derive first a sound axiomatic foundation of lottery dependent utility. Secondly, we develop a discontinuous variant of the model which can accommodate boundary effects and may lead to a lexicographic non-expected utility model. Both analyses are accompanied by considering some functional specifications which are in accordance with recent experimental results and may have significant applications in business and management science.     

Statistical decisions under ambiguity

This article provides unified axiomatic foundations for the most common optimality criteria in statistical decision theory. It considers a decision maker who faces a number of possible models of the world (possibly corresponding to true parameter values). Every model generates objective probabilities, and von Neumann–Morgenstern expected utility applies where these obtain, but no probabilities of models are given. This is the classic problem captured by Wald’s (Statistical decision functions, 1950) device of risk functions. In an Anscombe–Aumann environment, I characterize Bayesianism (as a backdrop), the statistical minimax principle, the Hurwicz criterion, minimax regret, and the “Pareto” preference ordering that rationalizes admissibility. Two interesting findings are that c-independence is not crucial in characterizing the minimax principle and that the axiom which picks minimax regret over maximin utility is von Neumann–Morgenstern independence.     

Bayesian decision theory,rule utilitarianism,and Arrow's impossibility theorem

The first part of this paper reexamines the logical foundations of Bayesian decision theory and argues that the Bayesian criterion of expected-utility maximization is the only decision criterion consistent with rationality. On the other hand, the Bayesian criterion, together with the Pareto optimality requirement, inescapably entails a utilitarian theory of morality. The next sections discuss the role both of cardinal utility and of cardinal interpersonal comparisons of utility in ethics. It is shown that the utilitarian welfare function satisfies all of Arrow's social choice postulates avoiding the celebrated impossibility theorem by making use of information which is unavailable in Arrow's original framework. Finally, rule utilitarianism is contrasted with act utilitarianism and judged to be preferable for the purposes of ethical theory.     

Mean-risk decision analysis

Some decision theorists criticize expected utility decision analysis and propose mean-risk decision analysis as a replacement. They claim that expected utility decision analysis neglects attitudes toward risk whereas mean-risk decision analysis accords these attitudes their proper status. However mean-risk decision analysis and expected utility decision analysis are not incompatible, and it is advantageous for decision theory to develop each in a way that complements the other. Here I present a mean-risk rule that governs preferences among options and options given states. This mean-risk rule complements an expected utility rule that takes the utility of an option-state pair as the utility of the option given the state. I argue for the mean-risk rule using principles concerning basic intrinsic desires. The rule is comparative, but the last section offers some suggestions for its quantitative development.I am grateful for comments from my colleague, Henry E. Kyburg, Jr.     

Choice under risk and the security factor: An axiomatic model

The particular attention paid by decision makers to the security level ensured by each decision under risk, which is responsible for the certainty effect, can be taken into account by weakening the independence and continuity axioms of expected utility theory. In the resulting model, preferences depend on: (i) the security level, (ii) the expected utility, offered by each decision. Choices are partially determined by security level comparison and completed by the maximization of a function, which express the existing tradeoffs between expected utility and security level, and is, at a given security level, an affine function of the expected utility. In the model, risk neutrality at a given security level implies risk aversion.     

Individual vs. couple behavior: an experimental investigation of risk preferences

In this article, we elicit both individuals’ and couples’ preferences assuming prospect theory (PT) as a general theoretical framework for decision under risk. Our experimental method, based on certainty equivalents, allows to infer measurements of utility and probability weighting at the individual level and at the couple level. Our main results are twofold. First, risk attitude for couples is compatible with PT and incorporates deviations from expected utility similar to those found in individual decision making. Second, couples’ attitudes towards risk are found to be consistent with a mix of individual attitudes, women being more influent on couples’ preferences at low probability levels.     

Local utility functions

In Machina's approach to generalised expected utility theory, decision makers maximise a choice functional which is smooth but not linear in the probabilities. When evaluating small changes, the choice functional can be approximated by the expectation of a local utility function. This local utility function is not however invariant under large changes in risk. This paper gives a simple explicit formula which can be used to write down the local utility functions of some common decision rules.     

A foundation of Bayesian statistics (How to deal with fears)

A rational statistical decision maker whose preferences satisfy Savage's axioms will minimize a Bayesian risk function: the expectation with respect to a revealed (or subjective) probability distribution of a loss (or negative utility) function over the consequences of the statistical decision problem. However, the nice expected utility form of the Bayesian risk criterion is nothing but a representation of special preferences. The subjective probability is defined together with the utility (or loss) function and it is not possible, in general, to use a given loss function - say a quadratic loss - and to elicit independently a subjective distribution.I construct the Bayesian risk criterion with a set of five axioms, each with a simple mathematical implication. This construction clearly shows that the subjective probability that is revealed by a decider's preferences is nothing but a (Radon) measure equivalent to a linear functional (the criterion). The functions on which the criterion operates are expected utilities in the von Neumann-Morgenstern sense. It then becomes clear that the subjective distribution cannot be eliciteda priori, independently of the utility function on consequences.However, if one considers a statistical decision problem by itself, losses, defined by a given loss function, become the consequences of the decisions. It can be imagined that experienced statisticians are used to dealing with different losses and are able to compare them (i.e. have preferences, or fears over a set of possible losses). Using suitable axioms over these preferences, one can represent them by a (linear) criterion: this criterion is the expectation of losses with respect to a (revealed) distribution. It must be noted that such a distribution is a measure and need not be a probability distribution.     

Subjectively weighted linear utility

An axiomatized theory of nonlinear utility and subjective probability is presented in which assessed probabilities are allowed to depend on the consequences associated with events. The representation includes the expected utility model as a special case, but can accommodate the Ellsberg paradox and other types of ambiguity sensitive behavior, while retaining familiar properties of subjective probability, such as additivity for disjoint events and multiplication of conditional probabilities. It is an extension, to the states model of decision making under uncertainty, of Chew's weighted linear utility representation for decision making under risk.     

Risk aversion elicitation: reconciling tractability and bias minimization

Risk attitude is known to be a key determinant of various economic and financial choices. Behavioral studies that aim to evaluate the role of risk attitudes in contexts of this type, therefore, require tools for measuring individual risk tolerance. Recent developments in decision theory provide such tools. However, the methods available can be time consuming. As a result, some practitioners might have an incentive to prefer “fast and frugal” methods to clean but more costly methods. In this article, we focus on a tractable procedure initially proposed by Holt and Laury (2002) to elicit risk attitude. We generalize this method to measure utility and risk aversion as follows. First, we allow measurement of probabilistic risk attitude through violations of expected utility due to probability weighting. Second, we use the outcome scale rather than the probability scale in the menu of choices. Third, we compare sure payoffs with lotteries instead of comparing non-degenerate lotteries. A within-subject experimental study illustrates the gains in tractability and bias minimization that can result from such an extension.     

Causality in the logic of decision

In recent years there has been an active debate between proponents of two different models of rational decision. One model is evidential decision theory, which is characterized by the fact that it holds the principle of maximizing expected utility to be appropriate whenever the states are probabilistically independent of the acts. The other model, causal decision theory, holds that the principle of maximizing expected utility is appropriate whenever the states are causally independent of the acts. The proponents of evidential decision theory include Richard Jeffrey and Ellery Eells, who claim that evidential decision theory has significant advantages over causal decision theory. In this paper I discuss the two main advantages which have been claimed for evidential decision theory, and show that in fact evidential decision theory does not possess either of these advantages.     

An Axiomatization of Cumulative Prospect Theory for Decision Under Risk

Cumulative prospect theory was introduced by Tversky and Kahneman so as to combine the empirical realism of their original prospect theory with the theoretical advantages of Quiggin's rank-dependent utility. Preference axiomatizations were provided in several papers. All those axiomatizations, however, only consider decision under uncertainty. No axiomatization has been provided as yet for decision under risk, i.e., the case in which given probabilities are transformed. Providing the latter is the purpose of this note. The resulting axiomatization is considerably simpler than that for uncertainty.     

Preferences for information on probabilities versus prizes: The role of risk-taking attitudes

Many real-world decisions entail choices between information on either probabilities or payoffs (i.e., prizes). Simplified versions of such decisions are examined to gain insight into preferences for different types of information as a function of risk-attitudes. General and simple decision rules are derived for cases where the utility function is concave (or convex) over the relevant payoff interval.The article further describes several experiments to test business students' intuitions concerning these optimal decision rules. In general, risk-taking attitudes did not correlate significantly with subjects' preferences for information, in violation of theorems regarding mean-preserving spreads of risk. Other tests, e.g., narrowing certain probability ranges, also resulted in preferences contrary to expected utility (EU) theory.     

Nontransitive preferences in decision theory

Intransitive preferences have been a topic of curiosity, study, and debate over the past 40 years. Many economists and decision theorists insist on transitivity as the cornerstone of rational choice, and even in behavioral decision theory intransitivities are often attributed to faulty experiments, random or sloppy choices, poor judgment, or unexamined biases. But others see intransitive preferences as potential truths of reasoned comparisons and propose representations of preferences that accommodate intransitivities. This article offers a partial survey of models for intransitive preferences in a variety of decisional contexts. These include economic consumer theory, multiattribute utility theory, game theory, preference between time streams, and decision making under risk and uncertainty. The survey is preceded by a discussion of issues that bear on the relevance and reasonableness of intransitivity.     

On the Inefficiency of Bang-Bang and Stop-Loss Portfolio Strategies

We show in this article that bang-bang portfolio strategies where the investor is alternatively 100% in equity and 100% in cash are dynamically inefficient. Our proof of this result is based on a simple second-order stochastic dominance (SSD) argument. It implies that this is true for any decision criterion that satisfies SSD, not necessarily expected utility. We also examine the stop-loss strategy in which the investor is 100 percent in equity as long as the value of the portfolio exceeds a lower limit where the investor switches to 100 percent in cash. Again, we show that this strategy is inefficient under second-order risk aversion. However, a slight modification of it–in which all wealth exceeding a minimum reserve is invested in equity–is shown to be an efficient dynamic portfolio strategy. This strategy is optimal for investors with a nondifferentiable utility function.     

Risk-adjusted martingales and the design of “indifference” gambles

In the probability literature, a martingale is often referred to as a “fair game.” A martingale investment is a stochastic sequence of wealth levels, whose expected value at any future stage is equal to the investor’s current wealth. In decision theory, a risk neutral investor would therefore be indifferent between holding on to a martingale investment, and receiving its payoff at any future stage, or giving it up and maintaining his current wealth. But a risk-averse decision maker would not be indifferent between a martingale investment and his current wealth level, since he values uncertain deals less than their mean. A risk seeking decision maker, on the other hand, would readily accept a martingale investment in exchange for his current wealth, and would repeat this investment any number of times. These ideas lead us to introduce the notion of a “risk-adjusted martingale”; a stochastic sequence of wealth levels that a rational decision maker with any attitude toward risk would value constantly with time, and would be indifferent between receiving its pay-off at any future stage, or giving it up and maintaining his current wealth level. We show how to construct such risk-adjusted investments for any decision maker with a continuous monotonic utility function. The fundamental result we derive is that a pay-off structure of an investment (i) is a risk-adjusted martingale and (ii) can be represented by a lattice if and only if the pay-off functions are invariant transformations of the given utility function.     

A note on Marshallian and von Neumann-Morgenstern utility

The curvature of a decision maker's utility function is often used to measure his risk preference. In order to comprehensively describe an individual's decision making behaviour, however, it would also seem desirable to measure the gain in utility from an increase in wealth or income before accounting for risk. If a small increase in wealth leads to a large utility gain, then it could be said that the individual's aspiration to achieve the wealth increase would be high. This aspiration, however, may be more than offset by the risk involved in obtaining this extra wealth and the individual's attitude towards risk. In the following paper it is shown how the marginal utility of Marshall can be used in a measure of aspiration with this measure then combined with the usual measure of risk preference to explain the shape of any individuals utility curve. Using these measures, a general utility curve for all income or wealth classes is postulated.The author would like to thank Professor I. Horowitz for providing the inspiration that led to his note. Any errors are the responsibility of the author.     

Elementary proof that mean–variance implies quadratic utility

An extensive literature overlapping economics, statistical decision theory and finance, contrasts expected utility [EU] with the more recent framework of mean–variance (MV). A basic proposition is that MV follows from EU under the assumption of quadratic utility. A less recognized proposition, first raised by Markowitz, is that MV is fully justified under EU, if and only if utility is quadratic. The existing proof of this proposition relies on an assumption from EU, described here as “Buridan’s axiom” after the French philosopher’s fable of the ass that starved out of indifference between two bales of hay. To satisfy this axiom, MV must represent not only “pure” strategies, but also their probability mixtures, as points in the (σ, μ) plane. Markowitz and others have argued that probability mixtures are represented sufficiently by (σ, μ) only under quadratic utility, and hence that MV, interpreted as a mathematical re-expression of EU, implies quadratic utility. We prove a stronger form of this theorem, not involving or contradicting Buridan’s axiom, nor any more fundamental axiom of utility theory.